https://projecteuler.net/problem=220

Let

*D*_{0}be the two-letter string "Fa". For n≥1, derive*D*_{n}from*D*_{n-1}by the string-rewriting rules:
"a" → "aRbFR"

"b" → "LFaLb"

"b" → "LFaLb"

Thus,

*D*_{0}= "Fa",*D*_{1}= "FaRbFR",*D*_{2}= "FaRbFRRLFaLbFR", and so on.
These strings can be interpreted as instructions to a computer graphics program, with "F" meaning "draw forward one unit", "L" meaning "turn left 90 degrees", "R" meaning "turn right 90 degrees", and "a" and "b" being ignored. The initial position of the computer cursor is (0,0), pointing up towards (0,1).

Then

*D*_{n}is an exotic drawing known as the*Heighway Dragon*of order*n*. For example,*D*_{10}is shown below; counting each "F" as one step, the highlighted spot at (18,16) is the position reached after 500 steps.
What is the position of the cursor after 10

Give your answer in the form

^{12}steps in*D*_{50}?Give your answer in the form

*x*,*y*with no spaces.Approaches

1. make this way using by phyton and turtle.

2. think more deeply to solve it.

3. I have to find pattern or math skill.

this is my phython coding.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 | __author__ = 'ddavid' import turtle a = 'aRbFR' b = 'LFaLb' s = 'Fa' wn = turtle.Screen() # Creates a playground for turtles tt = turtle.Turtle() # Create a turtle, assign to alex tt.speed(0) tt.left(90) tt.left(90) for i in range(14): s = s.replace('a', 't') s = s.replace('b', 'p') s = s.replace('t', a) s = s.replace('p', b) print(s) print('\n') if i == 9: for d in range(len(s)): print(s[d-1]) if s[d-1] == 'F': tt.forward(2) elif s[d-1] == 'L': tt.left(90) elif s[d-1] == 'R': tt.right(90) wn.mainloop() # Wait for user to close window |

if you run this code

you will see this result.

Have Fun~ enyoing euler problem site.

reference

https://en.wikipedia.org/wiki/Dragon_curve

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